Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1596 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,11)(7,12,10,8)(14,18)(15,21)(20,25)(23,27)(29,30,34,35)(31,38,36,32)(33,37,41,43)(39,42,44,40);(5,7,9,10)(6,8,11,12)(16,24)(20,25)(23,27)(26,28)(30,35)(31,36)(33,41)(40,42);(5,6,9,11)(7,8,10,12)(13,14,17,18)(15,22,21,19)(16,20,24,25)(23,28,27,26)(29,31,34,36)(30,32,35,38)(33,39,41,44)(37,40,43,42);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,15,17,21)(14,19,18,22)(16,23,24,27)(20,26,25,28)(29,32)(30,36)(31,35)(33,42)(34,38)(37,39)(40,41)(43,44);(5,8)(6,10)(7,11)(9,12)(13,16,17,24)(14,20,18,25)(15,23,21,27)(19,26,22,28)(29,33)(30,37)(31,39)(32,40)(34,41)(35,43)(36,44)(38,42)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=fgh,d^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5544554);